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			<h1 id="firstHeading" class="firstHeading">Downsampling</h1>
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<td class="mbox-text" style="">This article <b>needs additional <a href="http://en.wikipedia.org/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">citations</a> for <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a>.</b><br>
<small>Please help <a href="http://en.wikipedia.org/w/index.php?title=Downsampling&amp;action=edit" class="external text" rel="nofollow">improve this article</a> by adding <a href="http://en.wikipedia.org/wiki/Wikipedia:Identifying_reliable_sources" title="Wikipedia:Identifying reliable sources">reliable references</a>. Unsourced material may be <a href="http://en.wikipedia.org/wiki/Template:Citation_needed" title="Template:Citation needed">challenged</a> and <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. <i>(February 2010)</i></small></td>
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<p>In signal processing, <b>downsampling</b> (or "subsampling") is the process of <a href="http://en.wikipedia.org/wiki/Sample_rate_conversion" title="Sample rate conversion">reducing the sampling rate</a> of a <a href="http://en.wikipedia.org/wiki/Signal_%28information_theory%29" title="Signal (information theory)" class="mw-redirect">signal</a>. This is usually done to reduce the <a href="http://en.wikipedia.org/wiki/Data_signaling_rate" title="Data signaling rate">data rate</a> or the size of the data.</p>
<p>The downsampling factor (commonly denoted by <i>M</i>) is usually an 
integer or a rational fraction greater than unity. This factor 
multiplies the sampling time or, equivalently, divides the sampling 
rate. For example, if <a href="http://en.wikipedia.org/wiki/Red_Book_%28audio_Compact_Disc_standard%29" title="Red Book (audio Compact Disc standard)" class="mw-redirect">compact disc audio</a> at 44,100Hz is downsampled to 22,050Hz before broadcasting over <a href="http://en.wikipedia.org/wiki/FM_broadcasting" title="FM broadcasting">FM radio</a>, the <a href="http://en.wikipedia.org/wiki/Bit_rate" title="Bit rate">bit rate</a>
 is reduced in half, from 1,411,200 bit/s to 705,600 bit/s, assuming 
that each sample retains its size of 16 bits. The audio was therefore 
downsampled by a factor of 2.</p>
<table id="toc" class="toc">
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<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Maintaining_the_sampling_theorem_criterion"><span class="tocnumber">1</span> <span class="toctext">Maintaining the sampling theorem criterion</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Downsampling_process"><span class="tocnumber">2</span> <span class="toctext">Downsampling process</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Downsampling_by_integer_factor"><span class="tocnumber">2.1</span> <span class="toctext">Downsampling by integer factor</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Downsampling_by_rational_fraction"><span class="tocnumber">2.2</span> <span class="toctext">Downsampling by rational fraction</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-5"><a href="#See_also"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li>
</ul>
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<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Downsampling&amp;action=edit&amp;section=1" title="Edit section: Maintaining the sampling theorem criterion">edit</a>]</span> <span class="mw-headline" id="Maintaining_the_sampling_theorem_criterion">Maintaining the sampling theorem criterion</span></h2>
<p>Since downsampling reduces the sampling rate, we must be careful to make sure the <a href="http://en.wikipedia.org/wiki/Shannon-Nyquist_sampling_theorem" title="Shannon-Nyquist sampling theorem" class="mw-redirect">Shannon-Nyquist sampling theorem</a> criterion is maintained. If the sampling theorem is not satisfied then the resulting digital signal will have <a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">aliasing</a>. To ensure that the sampling theorem is satisfied, a <a href="http://en.wikipedia.org/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a> is used as an <a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filter</a> to reduce the bandwidth of the signal <i>before</i> the signal is downsampled; the overall process (low-pass filter, then downsample) is called <a href="http://en.wikipedia.org/wiki/Decimation_%28signal_processing%29" title="Decimation (signal processing)">decimation</a>.</p>
<p>Note that the anti-aliasing filter must be a low-pass filter in downsampling. This is different from <a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">sampling</a> a <a href="http://en.wikipedia.org/wiki/Continuous_signal" title="Continuous signal">continuous signal</a>, where either a low-pass filter or a <a href="http://en.wikipedia.org/wiki/Band-pass_filter" title="Band-pass filter">band-pass filter</a> may be used.</p>
<p><b>Remark</b>: A bandpass signal, i.e. a band-limited signal whose 
minimum frequency is different from zero, can be downsampled avoiding 
superposition of the spectra if certain conditions are satisfied (see 
e.g. <a href="http://spazioscuola.altervista.org/UndersamplingAR/UndersamplingARnv.htm" class="external autonumber" rel="nofollow">[1]</a>).</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Downsampling&amp;action=edit&amp;section=2" title="Edit section: Downsampling process">edit</a>]</span> <span class="mw-headline" id="Downsampling_process">Downsampling process</span></h2>
<p>Consider a <a href="http://en.wikipedia.org/wiki/Discrete_signal" title="Discrete signal">discrete signal</a> <span class="texhtml"><i>f</i>(<i>k</i>)</span> on a radian frequency <a href="http://en.wikipedia.org/wiki/Digital_frequency" title="Digital frequency">digital frequency</a> range.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Downsampling&amp;action=edit&amp;section=3" title="Edit section: Downsampling by integer factor">edit</a>]</span> <span class="mw-headline" id="Downsampling_by_integer_factor">Downsampling by integer factor</span></h3>
<p>Let <i>M</i> denote the downsampling factor.</p>
<ol>
<li>Filter the signal to ensure that the sampling theorem is satisfied. This filter should, theoretically, be the <a href="http://en.wikipedia.org/wiki/Sinc_filter" title="Sinc filter">sinc filter</a> with frequency cutoff at <img class="tex" alt="\frac{\pi}{M}" src="wikipedia-Downsampling_pliki/324e87b5ed56b5d7afb9cbd7a7db941e.png">. Let the filtered signal be denoted <span class="texhtml"><i>g</i>(<i>k</i>)</span>.</li>
<li>Reduce the data by picking out every <span class="texhtml"><i>M</i><sup><i>t</i><i>h</i></sup></span> sample: <span class="texhtml"><i>h</i>(<i>k</i>) = <i>g</i>(<i>M</i><i>k</i>)</span>. Data rate reduction occurs in this step.</li>
</ol>
<p>The first step calls for the use of a perfect low-pass filter, which 
is not implementable for real-time signals. When choosing a realizable 
low-pass filter this will have to be considered along with the aliasing 
effects it will have. Realizable low-pass filters have a "skirt", where 
the response diminishes from near unity to near zero. So in practice the
 cutoff frequency is placed far enough below the theoretical cutoff that
 the filter's skirt is contained below the theoretical cutoff.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Downsampling&amp;action=edit&amp;section=4" title="Edit section: Downsampling by rational fraction">edit</a>]</span> <span class="mw-headline" id="Downsampling_by_rational_fraction">Downsampling by rational fraction</span></h3>
<p>Let <i>M/L</i> denote the downsampling factor.</p>
<ol>
<li><a href="http://en.wikipedia.org/wiki/Upsampling" title="Upsampling">Upsample</a> by a factor of <i>L</i></li>
<li>Downsample by a factor of <i>M</i></li>
</ol>
<p>Note that a proper upsampling design requires an interpolation filter
 after increasing the data rate and that a proper downsampling design 
requires a filter before eliminating some samples. These two low-pass 
filters can be combined into a single filter.</p>
<p>Also note that these two steps are generally not reversible. 
Downsampling results in a loss of data and, if performed first, could 
result in data loss if there is any data filtered out by the 
downsampler's low-pass filter. Since both interpolation and 
anti-aliasing filters are low-pass filters, the filter with the smallest
 bandwidth is more restrictive and can therefore be used in place of 
both filters. Since the rational fraction <i>M/L</i> is greater than unity then <span class="texhtml"><i>L</i> &lt; <i>M</i></span> and the single low-pass filter should have cutoff at <span class="texhtml">π / <i>M</i></span>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Downsampling&amp;action=edit&amp;section=5" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Oversampling" title="Oversampling">Oversampling</a></li>
<li><a href="http://en.wikipedia.org/wiki/Sampling_%28information_theory%29" title="Sampling (information theory)" class="mw-redirect">Sampling (information theory)</a></li>
<li><a href="http://en.wikipedia.org/wiki/Signal_%28information_theory%29" title="Signal (information theory)" class="mw-redirect">Signal (information theory)</a></li>
<li><a href="http://en.wikipedia.org/wiki/Antialiasing" title="Antialiasing" class="mw-redirect">Antialiasing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Posterization" title="Posterization">Posterization</a></li>
</ul>
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<th style="" colspan="2" class="navbox-title"><span class="collapseButton">[<a href="#" id="collapseButton0">hide</a>]</span><span style="float:left;width:6em;text-align:left;"><span class="noprint plainlinks navbar" style=""><span style="white-space:nowrap;word-spacing:-.12em;"><a href="http://en.wikipedia.org/wiki/Template:DSP" title="Template:DSP"><span style=";;background:none transparent;border:none;font-size:100%;" title="View this template">v</span></a> <span style=";;background:none transparent;border:none;font-size:100%;"><b>·</b></span> <a href="http://en.wikipedia.org/wiki/Template_talk:DSP" title="Template talk:DSP"><span style=";;background:none transparent;border:none;font-size:100%;" title="Discuss this template">d</span></a> <span style=";;background:none transparent;border:none;font-size:100%;"><b>·</b></span> <a href="http://en.wikipedia.org/w/index.php?title=Template:DSP&amp;action=edit" class="external text" rel="nofollow"><span style=";;background:none transparent;border:none;font-size:100%;" title="Edit this template">e</span></a></span></span></span><span class="" style="font-size:110%;"><a href="http://en.wikipedia.org/wiki/Digital_signal_processing" title="Digital signal processing">Digital signal processing</a></span></th>
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<td class="navbox-group" style="">Theory</td>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Discrete-time_signal" title="Discrete-time signal" class="mw-redirect">discrete-time signal</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Sampling_theorem" title="Sampling theorem" class="mw-redirect">sampling theorem</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Estimation_theory" title="Estimation theory">estimation theory</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Detection_theory" title="Detection theory">detection theory</a></span></div>
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<td class="navbox-group" style="">Sub-fields</td>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Audio_signal_processing" title="Audio signal processing">audio signal processing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Control_engineering" title="Control engineering">control engineering</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Digital_image_processing" title="Digital image processing">digital image processing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Speech_processing" title="Speech processing">speech processing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Statistical_signal_processing" title="Statistical signal processing">statistical signal processing</a></span></div>
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<td class="navbox-group" style="">Techniques</td>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a> (DFT)&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT)&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Impulse_invariance" title="Impulse invariance">Impulse invariance</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Bilinear_transform" title="Bilinear transform">bilinear transform</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Pole%E2%80%93zero_mapping" title="Pole–zero mapping" class="mw-redirect">pole–zero mapping</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Z-transform" title="Z-transform">Z-transform</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Advanced_Z-transform" title="Advanced Z-transform">advanced Z-transform</a></span></div>
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<td class="navbox-group" style=""><a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">Sampling</a></td>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Oversampling" title="Oversampling">oversampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Undersampling" title="Undersampling">undersampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><strong class="selflink">downsampling</strong>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Upsampling" title="Upsampling">upsampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">aliasing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filter</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Sampling_rate" title="Sampling rate">sampling rate</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a>/<a href="http://en.wikipedia.org/wiki/Nyquist_frequency" title="Nyquist frequency">frequency</a></span></div>
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